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Conformal Window and Correlation Functions in Lattice Conformal - - PowerPoint PPT Presentation

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Y. Iwasaki U.Tsukuba and KEK 2012/12/04 SCGT2012 Conformal Window and Correlation Functions in Lattice Conformal QCD 2012 12 5 In Collaboration with K.-I. Ishikawa(U. Horoshima) Yu Nakayama(Caltech & IPMU) T.

slide-1
SLIDE 1

Conformal Window and Correlation Functions in Lattice Conformal QCD

  • Y. Iwasaki

U.Tsukuba and KEK 2012/12/04 SCGT2012

2012年12月5日水曜日

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SLIDE 2

In Collaboration with

K.-I. Ishikawa(U. Horoshima) Yu Nakayama(Caltech & IPMU)

  • T. Yoshie(U. Tsukuba)

2012年12月5日水曜日

slide-3
SLIDE 3

Plan of Talk

  • Briefly review our previous works
  • n the phase structure of the lattice SU(3) QCD.

Thereby clarify the reason why we conjecture that the conformal window is

  • Introduce the concept of “conformal theory with IR

cutoff”. Propose new predictions about the propagator of a meson. We verify that new numerical results satisfy our proposals for Nf=7 and Nf=16. (to be continued)

2012年12月5日水曜日

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SLIDE 4

Plan of Talk (Cont.)

  • Point out and verify that the propagator
  • f a meson at T/Tc >1 shows

the characteristics of “conformal theory with IR cutoff”.

2012年12月5日水曜日

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SLIDE 5

STAGES and TOOLS

  • Lattice gauge theory
  • one-plaqutte gauge action
  • Improved RG action: future plan
  • Wilson fermion action
  • Lattice size: Nx=Ny=Nz=N; Nt=r N
  • Lattice spacing:
  • PCAC quark mass:
  • G(t): propagators of mesons

2012年12月5日水曜日

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SLIDE 6

Strategy for Part 1

  • Investigate the phase structure

in terms of for

  • UV fixed point at
  • Find critical Nf for the Banks-Zaks IRFP
  • n the massless line starting from UVFP
  • Construct the field theory toward UVFP,

taking the limit and with constant

2012年12月5日水曜日

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SLIDE 7

Phase Diagram:

!"#$%#&'&#(

  • )&!"#$%#&'&#(

!

"

! #**+*,

$

#**+*,

$

Chiral transition on the massless line starting from the UVFP

The finite temperature phase transition in the quenched QCD transition and the chiral transition move toward larger beta, as N increases.

2012年12月5日水曜日

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SLIDE 8

Phase Diagram: ; N larger

!"#$%#&'&#(

  • !

"

!

!"#$%#&'&#(

  • !

"!)*)∞

#

$))*)+

%

As N increases, the green line becomes longer and in the limit N => infinity

  • nly the green part survives.

2012年12月5日水曜日

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SLIDE 9

Phase Diagram:

Complicated due to lack of chiral symmetry

  • 1. the massless line from the UVFP hits the bulk transition
  • 2. no massless line in the confining phase at strong coupling region

massless quark line only in the deconfining phase

!"#$%#&'&#(

  • )&!"#$%#&'&#(

!

"

! #

$

#

$

2012年12月5日水曜日

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SLIDE 10

Phase Diagram:

!"#$%#&'&#(

  • )&!"#$%#&'&#(

!

"

! #

$

#

$

!"#$%#&'&#(

  • !

"

! #

$

#

$

As N increases, massless quark line is, still,

  • nly in the deconfining phase

2012年12月5日水曜日

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SLIDE 11

What found

  • There are no green lines

(massless line in the confining phase)

for

  • Conformal window is
  • Indirect way to conclude this

2012年12月5日水曜日

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SLIDE 12

More direct way

  • Identify the IR fixed point

For small Nf, g* is in strong coupling region Only upper limit for g* ?

  • Find out characteristics of Conformal

theories <= this work

(End of part 1)

2012年12月5日水曜日

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SLIDE 13

Strategy for part 2

  • Define a continuous theory by continuum limit of

lattice theory, keeping L =finite or infinity

  • Introduce the concept “Conformal theories with

IR cutoff” in continuous theories

  • Then propose Conformal theories on a lattice

2012年12月5日水曜日

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SLIDE 14

Continuum limit

  • and , keeping constant
  • case 1: L=infinity IR cut-off ;
  • case 2: L=finite IR cut-off ;
  • A huge difference between case1 and

case2

2012年12月5日水曜日

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SLIDE 15

Case 1:

!"#$%#&'&#(

)*)+ ,-

  • ./

0&!"#$%#&1 '&#( !"#$"2'34

!))*)+

"

!

"

No physical quantities with physical dimensions conformal region exists only on the massless line massive region is confining phase

deconfining phase in strong coupling region is conjectured based on numerical simulations

2012年12月5日水曜日

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SLIDE 16

Case 1: Propagators of mesons

When When

UV fixed point IR fixed point scale invariant is a scale parameter for the transition region from UV to IR

2012年12月5日水曜日

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SLIDE 17

Case 2: Conformal theories with IR cutoff

!"#$%#&'&#(

)*)+ ,-

  • ./

!"#$"0'12

!))*)+

"

!

"

Physical quantities:

“Conformal theories with IR cutoff” region: see figure Boundary of the “conformal” region is given by

Propagators :

2012年12月5日水曜日

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SLIDE 18

Case 2: Conformal theories with IR cutoff (Cont.) and are t-dependent: evolve with RG transformation

2012年12月5日水曜日

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SLIDE 19

Conformal theories on the Lattice

  • Note: IR cutoff is inherent in numerical

simulations on a lattice:

  • Primary target of part 2 is to verify the

transition of meson propagators from an exponential damping form to a modified Yukawa-type, that is, an exponential form with power correction

2012年12月5日水曜日

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SLIDE 20

Size Dependence of Critical mass

When the lattice size is increased The critical mass is decreased If we keep the quark mass in the region Yukawa-type disappears Have to carefully choose the parameters to find the “Conformal region”

2012年12月5日水曜日

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SLIDE 21

Numerical Simulations

  • Algorithm: Blocked HMC for 2N and

RHMC for 1 : Nf=2N + 1

  • Computers:
  • U. Tsukuba: CCS HAPACS;

KEK: HITAC 16000

  • Nf=7, 16 and (Nf=2, 6)
  • Lattice size:
  • Statistics: 1,000 +1000 trajectories

2012年12月5日水曜日

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SLIDE 22

Parameters of Simulations

Masses are preliminary !

Nf7 K 0.1400 0.1446 0.1452 0.1459 0.1472 mq 0.22 0.084 0.062 0.045 0.006 mH(96) 0.66 0.33 0.33 0.20 mH(64) 0.68 0.46 0.42 0.41 0.41 mH(32) 0.74 0.74 0.74 Nf16 K 0.125 0.126 0.127 0.13 0.1315 0.13322 mq 0.25 0.22 0.19 0.1 0.055 0.003 mH(96) 0.30 0.27 0.32 mH(64) 0.54 0.54 0.49 0.43 0.38 0.38 mH(32)

2012年12月5日水曜日

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SLIDE 23

From now on, let me show you examples of Yukawa-type propagators for Nf=7 at beta=6.0 and Nf=16 at beta=11.5 detailed analyses will come later

2012年12月5日水曜日

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SLIDE 24

Nf7: mq=0.045: example of Yukawa-type

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 25

Nf=7: mq=0.22; example of exp!!-damp

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1400, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1400, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1400, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.65 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1400, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 26

Nf=7: mq=0.045 Yukawa-type fit[15:31]

0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel loc(t)-loc(0) fit[15:31]

2012年12月5日水曜日

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SLIDE 27

Nf16: mq=0.055; example of Yukawa-type

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 28

Nf16: mq=0.055: Yukawa-type fit[15:31]

0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.1315, Nf=16, 163x64, PS-channel loc(t)-loc(0) fit[15:31]

2012年12月5日水曜日

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SLIDE 29

More example :Nf=7; mq=0.084

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1446, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1446, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1446, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1446, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 30

More Nf=7; mq=0.062

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1452, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1452, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1452, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1452, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 31

More Nf=7; mq=0.0006

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1472, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1472, Nf=7, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1472, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=6.0, K=0.1472, Nf=7, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 32

Now Nf=16; mq=0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.130, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.130, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.130, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.130, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 33

More NF=16; mq=0.003

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.13322, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.13322, Nf=16, 163x64, PS-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.13322, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0) 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Beta=11.5, K=0.13322, Nf=16, 163x64, V-channel loc(t)-loc(0) loc(t)-dsmr(0) loc(t)-dwal(0)

2012年12月5日水曜日

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SLIDE 34

Verified the existence of “Conformal theories with IR cutoff” for Nf=7 and 16

Nf=7 for all lattice sizes at beta=6.0 Nf=16 for all lattice sizes at beta=11.5

Achieved the first target in part 2

2012年12月5日水曜日

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SLIDE 35

Second primary target in part 2

  • What kind of theory is defined ?
  • and reflect the dynamics
  • Investigate t-dependence of and

2012年12月5日水曜日

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SLIDE 36

: Nf=7; mq=0.084

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1446, Nf=16, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1446, Nf=16, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1446, Nf=16, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1446, Nf=16, 163x64, V-channel (loc-loc) win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 37

Compare with exp. damping Nf=7; K=0.1400(mq=0.22)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1400, Nf=7, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1400, Nf=7, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1400, Nf=7, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1400, Nf=7, 163x64, V-channel (loc-loc) win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 38

: Nf7; mq=0.062

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1452, Nf=7, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1452, Nf=7, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1452, Nf=7, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1452, Nf=7, 163x64, V-channel (loc-loc) win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 39

: Nf7; mq=0.045

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, V-channel (loc(t)-loc(0) win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, V-channel (loc(t)-loc(0) win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 40

: Nf7; mq=0.0006

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1472, Nf=16, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1472, Nf=16, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1472, Nf=16, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1472, Nf=16, 163x64, V-channel (loc-loc) win_size=3 win_size=5

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SLIDE 41

: Nf16; mq=0.084; N=16

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 163x64, V-channel (loc(t)-loc(0 win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 163x64, V-channel (loc(t)-loc(0 win_size=3 win_size=5

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SLIDE 42

0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

Nf16; mq=0.062; N=16

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SLIDE 43

: Nf16; mq=0.0006; N=16

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 163x64, V-channel (loc(t)-loc( win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 163x64, V-channel (loc(t)-loc( win_size=3 win_size=5

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SLIDE 44

: Nf16; mq=0.084; N24

0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 243x96, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.130, Nf=16, 243x96, V-channel (loc-loc) win_size=3 win_size=5

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SLIDE 45

: Nf16; mq=0.062; N24

0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 46

: Nf16; mq=0.0006; N24

0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13322, Nf=16, 243x96, V-channel (loc-loc win_size=3 win_size=5

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SLIDE 47

Observation of the results

  • Finite size effects are severe
  • Clear difference between Nf=7 and Nf=16
  • Nf７: plateau at t= 15 ~ 31 (16^3x64)
  • Nf16: shoulder at t= 12 ~ 24 (both sizes)
  • Compare the results with some models

for

2012年12月5日水曜日

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SLIDE 48

t-dependence of

  • In general, in the continuum limit
  • In the above derivation, assumed
  • In simulation results
  • To estimate in this case, we need a

model for

2012年12月5日水曜日

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SLIDE 49

Some models

  • a free Wilson quark and an anti-quark
  • meson unparticle model*
  • fermion unparticle model*

*: motivated by the soft-wall model in AdS/CFT correspondence : meson operator

2012年12月5日水曜日

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SLIDE 50

Model calculations

In case Free case: Meson unparticle case: Fermion unparticle case: , for

2012年12月5日水曜日

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SLIDE 51

Interpretation of Results

  • Nf=7 is close to the meson unparticle model
  • Nf=16 is close to the fermion unparticle model

plateau at shoulder at

consistent with 2-loop results:

(End of part 2)

2012年12月5日水曜日

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SLIDE 52

Strategy for Part 3

  • Note that QCD at high temperature is the

theory with IR cutoff

  • Apply a similar idea of Part 2 to this case
  • Derive physical implications

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SLIDE 53

Running Coupling Constant at T In UV region, the theory is asymptotic free, therefore perturbative RG is applicable the coupling constant is universal Define a running coupling constant by any method such as Wilson method

cf: Kaczmarek(2004) et. al

  • n the line

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SLIDE 54

Running Coupling Constant at T (Cont.)

  • In IR region, the running constant

may be quite different from since the IR cutoff is finite; 1/T

  • When T/Tc >1, cannot be arbitrarily

large, since the quark is not confined

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SLIDE 55

!!" !!#$"#" "#%&%"$

'

!!" !!#$"#" "#%&%"$

!!" !!#$"#" "#%&&%"$

Beta function for T > Tc

As far as , the beta-function is negative all through When , but the beta function changes the sign from negative to positive at large When T increases further, it will change the sign at medium strong g When it will change the sign at small

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SLIDE 56

Nf=2; T~ 2 Tc

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

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SLIDE 57

T~100Tc

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

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SLIDE 58

T ~ 10^5Tc

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=15.0, K=0.130, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=15.0, K=0.130, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=15.0, K=0.130, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=15.0, K=0.130, Nf=2, 163x64, V-channel (loc-loc) win_size=3 win_size=5

2012年12月5日水曜日

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SLIDE 59

Similarity between large Nf with and small Nf at T/Tc >1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=10.0, K=0.135, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 Fit Mass Fit Pow 0.1 0.2 0.3 0.4 0.5 0.6 0.7 4 8 12 16 20 24 28 32 36 40 44 48 Fit Mass t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 4 8 12 16 20 24 28 32 36 40 44 48 Fit Pow t of Fit Range [t,t+win_size-1] Beta=11.5, K=0.13150, Nf=16, 243x96, PS-channel (loc-loc) win_size=3 win_size=5 Fit Mass Fit Pow

Nf16; with Nf2; T= 100Tc

2012年12月5日水曜日

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SLIDE 60

Similarity between large Nf with and small Nf at T/Tc >1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5

  • 0.5

0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.5, K=0.146, Nf=2, 163x64, PS-channel (loc-loc) win_size=3 win_size=5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Mass t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 Fit Pow t of Fit Range [t,t+win_size-1] Beta=6.0, K=0.1459, Nf=7, 163x64, PS-channel (loc(t)-loc(0)) win_size=3 win_size=5

Nf7; with Nf2; T = 2 Tc

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SLIDE 61

Long standing important issues

  • Free energy of quark-gluon plasma state

does not reach that of the Stefan-Boltzman idea gas state even at T/Tc=100

  • Wave function of “meson” just above Tc can

be obtained, although quarks are deconfined

  • The order of chiral phase transition in Nf=2

case: 1st or 2nd ? : UA(1) symmetry ?

Pisarski and Wilczek(1984), iwasaki et. al(1997);

  • S. Aoki et. al(2012)

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SLIDE 62

Solutions

  • quarks and gluons are not free particles

When T ~ Tc, meson unparticles When T >> Tc, fermion unparticles

  • meson unparticles are similar to meson

particles in some aspects

  • G(t) is not analytic in terms of mq and mH

(End of part 3)

2012年12月5日水曜日

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SLIDE 63

Conclusions

  • “Conformal Theories with IR cutoff” are

satisfactorily verified in the cases of Nf=7 and Nf=16

  • The assertion that the Conformal Window

is , is thereby strengthened

  • “Conformal Theories with IR cutoff” are also

verified in the case of T/Tc >1 in Nf=2 and Nf=6

  • IR cutoff is inherent with simulations on a lattice

and QCD at high temperatures (to be continued)

2012年12月5日水曜日

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SLIDE 64

Conclusions (Cont.)

  • “Nf=7” and “T~Tc” are similar to each
  • ther, and are consistent with meson

unparticle model

  • “Nf=16” and “T>>Tc” are similar to each
  • ther, and are consistent with fermion

unparticle model

  • Physics implications should be deepened
  • A lot of things should be done

2012年12月5日水曜日